Area under Curve/Examples/sin x from 0 to pi
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Example of Use of Area under Curve
The area bounded by the curve $y = \sin x$ and the $x$-axis between $x = 0$ and $x = \pi$ is $2$.
Proof
Let $\AA$ be the area in question.
From Area under Curve we need to evaluate the definite integral:
- $\AA = \ds \int_0^\pi \sin x \rd x$
So:
\(\ds \AA\) | \(=\) | \(\ds \int_0^\pi \sin x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \bigintlimits {-\cos x} 0 \pi\) | Primitive of Sine Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-\cos \pi} - \paren {-\cos 0}\) | Definition of Definite Integral | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-\paren {-1} } - \paren {-1}\) | Cosine of Straight Angle, Cosine of $0 \degrees$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 2\) | evaluation |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Exercises $\text {XV}$: $3$